Decay of Axisymmetric Solutions of the Wave Equation on Extreme Kerr Backgrounds

Abstract: We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of S. We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T, and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N. Finally, we prove uniform pointwise boundedness and power-law decay for solutions to the wave equation. Our estimates hold up to and including the event horizon. We remark that these results do not yield decay for the derivatives transversal to the horizon, and this is suggestive that these derivatives may satisfy instability properties analogous to those shown in our previous work on extreme Reissner-Nordstrom backgrounds.